mars

Multivariate Adaptive Regression Spline (MARS)

Specification

• Alias: None

• Arguments: None

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Optional		max_bases	Maximum number of MARS bases
Optional		interpolation	MARS model interpolation type
Optional		export_model	Exports surrogate model in user-specified format(s)
Optional		import_model	Import surrogate model from archive file

Description

This surface fitting method uses multivariate adaptive regression splines from the MARS3.5 package [Fri91] developed at Stanford University.

The MARS reference material does not indicate the minimum number of data points that are needed to create a MARS surface model. However, in practice it has been found that at least $n_{c_{quad}}$ , and sometimes as many as 2 to 4 times $n_{c_{quad}}$ , data points are needed to keep the MARS software from terminating. Provided that sufficient data samples can be obtained, MARS surface models can be useful in SBO and OUU applications, as well as in the prediction of global trends throughout the parameter space.

Known Issue: When using discrete variables, there have been sometimes significant differences in surrogate behavior observed across computing platforms in some cases. The cause has not yet been fully diagnosed and is currently under investigation. In addition, guidance on appropriate construction and use of surrogates with discrete variables is under development. In the meantime, users should therefore be aware that there is a risk of inaccurate results when using surrogates with discrete variables.

Theory

The form of the MARS model is based on the following expression:

$$\hat{f}(\mathbf{x})=\sum _{m=1}^M{a}_m{B}_m(\mathbf{x})$$

where the $a_m$ are the coefficients of the truncated power basis functions $B_m$ , and $M$ is the number of basis functions. The MARS software partitions the parameter space into subregions, and then applies forward and backward regression methods to create a local surface model in each subregion. The result is that each subregion contains its own basis functions and coefficients, and the subregions are joined together to produce a smooth, $C^2$ -continuous surface model.

MARS is a nonparametric surface fitting method and can represent complex multimodal data trends. The regression component of MARS generates a surface model that is not guaranteed to pass through all of the response data values. Thus, like the quadratic polynomial model, it provides some smoothing of the data.